C₂×#13657 #512776

order	512
nilpotency class	5
isoclinism	isoclinic class of #13657
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13657

Subgroups

	subgroup	G/H
center, socle	C₂²	#2177
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

418 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 66 points:

(1,2);(3,4)(5,17,20,51)(6,14)(7,25)(8,31)(9,34)(10,16)(11,19,37,58)(12,23)(13,28)(15,33)(18,53,55,66)(21,60,35,52)(22,61)(24,46)(26,45,54,50)(27,43)(29,40)(30,48)(32,49,57,41)(36,64,38,65)(39,63,47,59)(42,62)(44,56);(3,5)(4,11)(6,18)(7,26)(8,32)(9,35)(10,20)(12,36)(13,41)(14,45)(15,47)(16,37)(17,48)(19,56)(21,33)(22,59)(23,57)(24,55)(25,51)(27,63)(28,58)(29,65)(30,54)(31,38)(34,39)(40,60)(42,52)(43,53)(44,49)(46,50)(61,66)(62,64);(3,6,9,23,10,24,33,8)(4,12,15,14,16,31,34,46)(5,18,35,57,20,55,21,32)(7,27,29,28,30,22,62,56)(11,36,47,45,37,38,39,50)(13,42,43,48,44,40,61,25)(17,49,60,66,51,41,52,53)(19,26,63,65,58,54,59,64)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#512778, C₂×#151105, C₂×#512779, …
maximal normal subgroup of	C₂×#512778, C₂×#151105, C₂×#512779, …
maximal quotient of	C₂×#152000, C₂×#151972, C₂×#151083, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419972);